A Survey of Probability Concepts
Chapter 05
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
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LEARNING OBJECTIVES
LO 5-1 Explain the terms experiment, event, and outcome.
LO 5-2 Identify and apply the appropriate approach to assigning probabilities.
LO 5-3 Calculate probabilities using the rules of addition.
LO 5-4 Define the term joint probability.
LO 5-5 Calculate probabilities using the rules of multiplication.
LO 5-6 Define the term conditional probability.
LO 5-7 Compute probabilities using a contingency table.
LO 5-8 Determine the number of outcomes using the appropriate principle of counting.
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Probability
PROBABILITY A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.
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Experiment, Outcome, and Event
- An experiment is a process that leads to the occurrence of one, and only one, of several possible observations.
- An outcome is the particular result of an experiment.
- An event is the collection of one or more outcomes of an experiment.
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LO 5-1 Explain the terms experiment, event, and outcome
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Ways of Assigning Probability
Three ways of assigning probability:
1. CLASSICAL PROBABILITY
Based on the assumption that the outcomes of an experiment are equally likely.
2. EMPIRICAL PROBABILITY
The probability of an event happening is the fraction of the time similar events happened in the past.
3. SUBJECTIVE CONCEPT OF PROBABILITY
The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.
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LO 5-2 Identify and apply the appropriate approach to assigning probabilities.
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Classical Probability
Consider an experiment of rolling a six-sided die. What is the probability of the event that “an even number of spots appears face up”?
The possible outcomes are:
There are three “favorable” outcomes (a two, a four, and a six) in the collection of six equally likely possible outcomes.
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LO 5-2
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Mutually Exclusive Events and Collectively Exhaustive Events
- Events are mutually exclusive if the occurrence of any one event means that none of the others can occur at the same time.
- Events are independent if the occurrence of one event does not affect the occurrence of another. Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.
- Events are collectively exhaustive if at least one of the events must occur when an experiment is conducted.
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LO 5-2
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Empirical Probability
The empirical approach to probability is based on what is called the Law of Large Numbers.
Key to establishing probabilities empirically: a larger number of observations provides a more accurate estimate of the probability.
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LO 5-2
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The Law of Large Numbers
Key to establishing probabilities empirically: a larger number of observations provides a more accurate estimate of the probability.
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LO 5-2
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The Law of Large Numbers
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LO 5-2
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Empirical Probability ̶ Example
On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in 123 space missions for NASA. On the basis of this information, what is the probability that a future mission is successfully completed?
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LO 5-2
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Subjective Probability ̶ Example
- Use subjective probability if there is little or no past experience or information on which to base a probability.
- Illustrations of subjective probability are:
1. Estimating the likelihood the New England Patriots will play in the Super Bowl next year.
2. Estimating the likelihood you will be married before the age of 30.
3. Estimating the likelihood the U.S. budget deficit will be reduced by half in the next 10 years.
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LO 5-2
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Summary of Types of Probability
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LO 5-2
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Rules of Addition for Computing Probabilities
Rules of Addition
- Special Rule of Addition - If two events A and B are mutually exclusive, the probability of one or the other event’s occurring equals the sum of their probabilities.
P(A or B) = P(A) + P(B)
- The General Rule of Addition - If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:
P(A or B) = P(A) + P(B) ̶ P(A and B)
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LO 5-3 Calculate probabilities using rules of addition.
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Addition Rule – Mutually Exclusive Events Example
An automatic Shaw machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. A check of 4,000 packages filled in the past month revealed:
What is the probability that a particular package will be either underweight or overweight?
P(A or C) = P(A) + P(C) = .025 + .075 = .10
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LO 5-3
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Addition Rule – Not Mutually Exclusive Events Example
What is the probability that a card chosen at random from a standard deck of cards will be either a king or a heart?
P(A or B) = P(A) + P(B) ̶ P(A and B)
= 4/52 + 13/52 ̶ 1/52
= 16/52, or .3077
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LO 5-3
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The Complement Rule
The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1.
P(A) + P(~A) = 1
or P(A) = 1 ̶ P(~A).
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LO 5-3
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The Complement Rule ̶ Example
An automatic Shaw machine fills plastic bags with a mixture of beans, broccoli, and other vegetables. Most of the bags contain the correct weight, but because of the variation in the size of the beans and other vegetables, a package might be underweight or overweight. Use the complement rule to show the probability of a satisfactory bag is 0.900.
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LO 5-3
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The Complement Rule – Example
P(B) = 1 – P(~B)
= 1 – P(A or C)
= 1 – [P(A) + P(C)]
= 1 – [.025 + .075]
= 1 – .10
= .90
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LO 5-3
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The General Rule of Addition
The Venn Diagram shows the result of a survey of 200 tourists who visited Florida during the year. The survey revealed that 120 went to Disney World, 100 went to Busch Gardens, and 60 visited both.
What is the probability a selected person visited either Disney World or Busch Gardens?
P(Disney or Busch) = P(Disney) + P(Busch) – P(both Disney and Busch)
= 120/200 + 100/200 – 60/200
= .60 + .50 – .80
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LO 5-3
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Joint Probability – Venn Diagram
JOINT PROBABILITY A probability that measures the likelihood two or more events will happen concurrently.
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LO 5-4 Define the term joint probability.
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Rules of Addition
- Independent Events
P(A or B) = P(A) + P(B)
- Dependent Events
P(A or B) = P(A) + P(B) – P(A and B)
Joint Probability – Independent versus Dependent Events
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LO 5-4
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Special Rule of Multiplication
- The special rule of multiplication requires that two events A and B are independent.
- Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.
- This rule is written: P(A and B) = P(A)P(B)
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LO 5-5 Calculate probabilities using the rules of multiplication.
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Multiplication Rule – Example
A survey by the American Automobile association (AAA) revealed 60 percent of its members made airline reservations last year. Two members are selected at random. Since the number of AAA members is very large, we can assume that R1 and R2 are independent.
What is the probability both made airline reservations last year?
Solution:
The probability the first member made an airline reservation last year is .60, written as P(R1) = .60.
The probability that the second member selected made a reservation is also .60, so P(R2) = .60.
Since the number of AAA members is very large, it can be assumed that R1 and R2 are independent.
P(R1 and R2) = P(R1)P(R2) = (.60)(.60) = .36
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LO 5-5
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Conditional Probability
- A conditional probability is the probability of a particular event occurring, given that another event has occurred.
- The probability of the event A occurring given that the event B has occurred is written P(A|B).
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LO 5-6 Define the term conditional probability.
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General Multiplication Rule
The general rule of multiplication is used to find the joint probability that two independent events will occur.
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LO 5-6
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General Multiplication Rule – Example
Probability Expression of the Question:
P(W1and W2) = ?
W1and W2 are dependent
A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.
What is the likelihood both shirts selected are white?
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LO 5-6
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General Multiplication Rule – Example
- First Day: P(W1) = 9/12
- Second Day: P(W2 | W1) = 8/11.
- To determine the probability of selecting 2 white, we use formula:
P(AB) = P(A) P(B|A)
- P(W1 and W2) = P(W1)P(W2 |W1) = (9/12)(8/11) = 0.55
A golfer has 12 golf shirts in his closet. Suppose 9 of these shirts are white and the others blue. He gets dressed in the dark, so he just grabs a shirt and puts it on. He plays golf two days in a row and does not do laundry.
What is the likelihood both shirts selected are white?
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LO 5-6
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Contingency Tables
A CONTINGENCY TABLE is a table used to classify sample observations according to two or more identifiable characteristics.
E.g., A survey of 150 adults classified each as to gender and the number of movies attended last month. Each respondent is classified according to two criteria—the number of movies attended and gender.
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LO 5-7 Compute probabilities using a contingency table.
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Contingency Tables – Example
A sample of executives were surveyed about their loyalty to their company. One of the questions was, “If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position?” The responses of the 200 executives in the survey were cross-classified with their length of service with the company.
What is the probability of randomly selecting an executive who is loyal to the company (would remain) and who has more than 10 years of service?
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LO 5-7
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Contingency Tables – Example
Event A1 - if a randomly selected executive will remain with the company despite an equal or slightly better offer from another company. Since there are 120 executives out of the 200 in the survey who would remain with the company
P(A1) = 120/200, or .60
Event B4 - if a randomly selected executive has more than 10 years of service with the company. Thus, P(B4| A1) is the conditional probability that an executive with more than 10 years of service would remain with the company. Of the 120 executives who would remain, 75 have more than 10 years of service, so
P(B4| A1) = 75/120
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LO 5-7
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Counting Rules – Multiplication
The multiplication formula indicates that if there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.
Example: Dr. Delong has 10 shirts and 8 ties. How many shirt and tie outfits does he have?
(10)(8) = 80
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LO 5-8 Determine the number of outcomes using the appropriate principle of counting.
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Counting Rules – Multiplication: Example
An automobile dealer wants to advertise that for $29,999 you can buy a convertible, a two-door sedan, or a four-door model with your choice of either wire wheel covers or solid wheel covers. How many different arrangements of models and wheel covers can the dealer offer?
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LO 5-8
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Counting Rules – Permutation
A permutation is any arrangement of r objects selected from n possible objects. The order of arrangement is important in permutations.
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LO 5-8
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Permutation Example
A group of three electronic parts are to be assembled in any order. How many different ways can they be assembled?
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LO 5-8
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Permutation – Another Example
Betts Machine Shop Inc. has eight screw machines but only three spaces available in the production area for the machines. In how many different ways can the eight machines be arranged in the three spaces available?
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LO 5-8
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Counting Rules – Combination
A combination is the number of ways to choose r objects from a group of n objects without regard to order.
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LO 5-8
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Combination Example
The marketing department has been given the assignment of designing color codes for the 42 different lines of compact disks sold by Goody Records. Three colors are to be used on each CD, but a combination of three colors used for one CD cannot be rearranged and used to identify a different CD. This means that if green, yellow, and violet were used to identify one line, then yellow, green, and violet (or any other combination of these three colors) cannot be used to identify another line. Would seven colors taken three at a time be adequate to color-code the 42 lines?
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LO 5-8
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Combination Example
Would 7 colors taken 3 at a time be adequate to color-code the 42 lines?
n = 7 colors to choose from
r = 3 colors to choose each time
Order of color selection is not important
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LO 5-8
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Combination – Another Example
There are 12 players on the Carolina Forest High School basketball team. Coach Thompson must pick 5 players among the 12 on the team to comprise the starting lineup. How many different groups are possible?
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LO 5-8
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Discrete Probability Distributions
Chapter 06
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
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The study of probability begins with understanding the two basic classifications of a random variable–discrete and continuous. In order to apply the correct formula for solving a probability question, one must be able to know whether the variable is discrete or continuous.
LEARNING OBJECTIVES
LO 6-1 Identify the characteristics of a probability distribution.
LO 6-2 Distinguish between a discrete and a continuous random variable.
LO 6-3 Compute the mean of a probability distribution.
LO 6-4 Compute the variance and standard deviation of a probability distribution.
LO 6-5 Describe and compute probabilities for a binomial distribution.
LO 6-6 Describe and compute probabilities for a Poisson distribution.
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In this chapter, we investigate the commonly occurring discrete probability distributions–namely, the uniform, binomial, and poisson. We explore examples demonstrating situations involving these discrete distributions.
We are going to define the terms probability distribution and random variable. We will learn the distinction between discrete and continuous probability distributions. For a discrete probability distribution, we will learn how to calculate its mean, variance, and standard deviation. We’re also going to describe the characteristics of and compute probabilities using the binomial and the Poisson probability distributions.
What Is a Probability Distribution?
Experiment:
Toss a coin three times. Observe the number of times heads appears.
The possible results are:
Zero heads
One heads
Two heads
Three heads
What is the probability distribution for the number of heads?
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PROBABILITY DISTRIBUTION A listing of all the outcomes of an experiment and the probability associated with each outcome.
LO 6-1 Identify the characteristics of a probability distribution.
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A probability distribution is a listing of all the outcomes of an experiment and the probability associated with each outcome. Probability distributions are characterized by the following: the probability of a particular outcome, usually denoted by P of x, is between zero and one inclusive. The outcomes, denoted by x, are mutually exclusive; and the list is exhaustive so that the sum of the probabilities of the various outcomes is equal to 1.
For example, consider an experiment involving tossing a coin three times. We let x represent the number of heads observed in three tosses. The exhaustive list for x list is zero heads, one heads, two heads, and three heads. Altogether, there are eight possible unique combinations of how the three coins will appear, as shown in the table in the slide. The probability of observing zero heads is one of eight outcomes. Since the list of x is exhaustive, the sum of their probabilities is 1.
Characteristics of a Probability Distribution
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- The probability of a particular outcome is between 0 and 1 inclusive.
2. The outcomes are mutually exclusive events.
3. If the list is collectively exhaustive, the sum of the probabilities of the various events is 1.
LO 6-1
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Discrete probability distributions have the following characteristics:
One, the probability of a particular outcome is between 0 and 1 inclusive.
Two, the outcomes are mutually exclusive events.
Three, the list is exhaustive. So the sum of the probabilities of the various events is equal to 1.
Probability Distribution of the Number of Heads Observed in 3 Tosses of a Coin
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LO 6-1
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In an earlier slide, we considered an experiment involving tossing a coin three times. We were interested in observing the number of heads in three tosses. We denoted this random number as x. The table on the slide shows the exhaustive list for x : zero heads, one heads, two heads, and three heads. Altogether, there are eight possible unique combinations of how the three coins will appear, shown in the table in the slide. The probability of observing zero heads is one of eight outcomes, or 0.125. Because x is listed exhaustively, the sum of their probabilities is equal to one. The figure on the right of the table graphs the probabilities of the respective outcomes. The resulting graph outlines the shape of the probability distribution of this coin tossing experiment.
Random Variables
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RANDOM VARIABLE A quantity resulting from an experiment that, by chance, can assume different values.
LO 6-1
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We define random variable. A random variable is a value or number that results from an experiment that by chance can take on different values. We usually use the letter x to denote a random variable.
In the coin tossing experiment discussed in the previous slide, the random variable of the experiment is the number of heads that will appear in three tosses of a coin. The possible values are 0, 1, 2, or 3.
Types of Random Variables
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DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.
CONTINUOUS RANDOM VARIABLE Can assume an infinite number of values within a given range. It is usually the result of some type of measurement.
LO 6-2 Distinguish between a discrete and a continuous random variable.
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Earlier, we defined a random variable as an observation, a quantity, or a number resulting from an experiment that by chance can assume different values. Random variables are classified as either discrete or continuous. A random number that can assume only certain clearly separated values, usually the result of counting something, is known as a discrete random number.
Random numbers that can assume an infinite number of values within a range, usually the result of some type of measurement, are known as continuous random numbers.
Discrete Random Variables
EXAMPLES
The number of students in a class.
The number of children in a family.
The number of cars entering a carwash in an hour.
Number of home mortgages approved by Coastal Federal Bank last week.
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DISCRETE RANDOM VARIABLE A random variable that can assume only certain clearly separated values. It is usually the result of counting something.
LO 6-2
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Continuous Random Variables
EXAMPLES
- The length of each song on the latest Tim McGraw album.
- The weight of each student in this class.
- The temperature outside as you are reading this book.
- The amount of money earned by each of the more than 750 players currently on Major League Baseball team rosters.
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CONTINUOUS RANDOM VARIABLE Can assume an infinite number of values within a given range. It is usually the result of some type of measurement.
LO 6-2
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The Mean of a Probability Distribution
MEAN
- A typical value used to represent the central location of a probability distribution.
- The mean of a probability distribution is also referred to as its expected value.
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LO 6-3 Compute the mean of a probability distribution.
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Mean, Variance, and Standard
Deviation of a Probability Distribution – Example
John Ragsdale sells new cars for Pelican Ford. He has developed the following probability distribution for the number of cars he expects to sell on a particular Saturday.
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LO 6-3
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Mean of a Probability Distribution –
Example
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LO3
LO 6-3
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The Variance and Standard
Deviation of a Probability Distribution
- Measure the amount of spread in a distribution
- Computational steps:
1. Compute the mean
2. Subtract the mean from each value, and square this difference.
3. Multiply each squared difference by its probability.
4. Sum the resulting products to arrive at the variance.
5. Take the positive square root of the variance to obtain the standard deviation.
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LO 6-4 Compute the variance and standard deviation of a probability distribution.
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Variance and Standard
Deviation of a Probability Distribution – Example
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LO 6-4
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Binomial Probability Distribution
- A widely occurring discrete probability distribution
- Characteristics of a binomial probability distribution
There are only two possible outcomes of a particular trial of an experiment.
The outcomes are mutually exclusive.
The random variable is the result of counts.
Each trial is independent of any other trial.
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LO 6-5 Describe and compute probabilities for a binomial distribution.
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Binomial Probability Experiment
1. An outcome on each trial of an experiment is classified into one of two mutually exclusive categories—a success or a failure.
- The random variable counts the number of successes in a fixed number of trials.
- The probability of success and failure stay the same for each trial.
- The trials are independent, meaning that the outcome of one trial does not affect the outcome of any other trial.
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LO 6-5
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Binomial Probability Formula
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LO 6-5
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Binomial Probability – Example
There are five flights daily from Pittsburgh via US Airways into the Bradford Regional Airport in PA. Suppose the probability that any flight arrives late is .20.
What is the probability that none of the flights are late today?
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LO 6-5
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Binomial Probability – Excel
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LO 6-5
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Binomial Distribution – Mean and Variance
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LO 6-5
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For the example regarding the number of late flights, recall that = .20 and n = 5.
What is the average number of late flights?
What is the variance of the number of late flights?
Binomial Distribution – Mean and Variance: Example
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LO 6-5
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Binomial Distribution – Mean and Variance: Another Solution
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LO 6-5
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Binomial Distribution – Table
In a region of a country, five percent of all cell phone calls are dropped. What is the probability that out of six randomly selected calls, none were dropped?
Given Data:
n = 6 (sample size)
π = 0.05 (probability of success – dropped call)
x = 0 (number of dropped calls)
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LO 6-5
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Binomial Distribution – Table
Given Data:
n = 6, π = 0.05, x = 0
Find P(x = 0) =?
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LO 6-5
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Binomial Distribution – Table
Given Data:
n = 6, π = 0.05, x = 0
What is the probability that out of six randomly selected calls exactly one, exactly two, exactly three, exactly four, exactly five, or exactly six are dropped calls?
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LO 6-5
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Binomial Distribution – MegaStat
In a region of a country, five percent of all cell phone calls are dropped. What is the probability that out of six randomly selected calls, …
None will be dropped?
Exactly one?
Exactly two?
Exactly three?
Exactly four?
Exactly five?
Exactly six out of six?
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LO 6-5
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Binomial – Shapes for Varying (n constant)
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LO 6-5
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Binomial – Shapes for Varying n ( constant)
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LO 6-5
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Binomial Probability Distributions – Example
A study by the Illinois Department of Transportation concluded that 76.2 percent of front seat occupants used seat belts. A sample of 12 vehicles is selected.
What is the probability the front seat occupants in exactly 7 of the 12 vehicles are wearing seat belts?
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LO 6-5
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Binomial Probability Distributions – Example
Given Data:
n = 12 vehicles (sample size)
π = 0.762 (probability of success – wearing seatbelt)
x = 7 (front seat occupants wearing seatbelts)
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LO 6-5
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Binomial Probability Distributions – Example
Given Data:
n = 12 vehicles
π = 0.762 (proportion wearing seatbelt)
What is the probability the front seat occupants in at least 7 of the 12 vehicles are wearing seat belts?
P(x ≥ 7) = ?
P(x =7,8,9,10,11,12) = ?
P(x ≥ 7) = 0.9562
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LO 6-5
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Cumulative Binomial Probability Distributions – Excel
=binomdist(6,12,0.762,0)
=1- binomdist(6,12,0.762,1)
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LO 6-5
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Poisson Probability Distribution
The Poisson probability distribution describes the number of times some event occurs during a specified interval. The interval may be time, distance, area, or volume.
Assumptions of the Poisson Distribution
- The probability is proportional to the length of the interval.
- The intervals are independent.
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LO 6-6 Describe and compute probabilities for a Poisson distribution.
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Poisson Probability Distribution
The Poisson probability distribution is characterized by the number of times an event happens during some interval or continuum.
Examples include:
• The number of misspelled words per page in a newspaper.
• The number of calls per hour received by Dyson Vacuum Cleaner Company.
• The number of vehicles sold per day at Hyatt Buick GMC in Durham, North Carolina.
• The number of goals scored in a college soccer game.
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LO 6-6
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Poisson Probability Distribution
The Poisson distribution can be described mathematically using the formula:
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LO 6-6
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Poisson Probability Distribution
- The mean number of successes μ can be determined in Poisson situations by n, where n is the number of trials and the probability of a success.
- The variance of the Poisson distribution is also equal to n.
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LO 6-6
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Poisson Probability Distribution – Example
Assume baggage is rarely lost by Northeast Airlines. Suppose a random sample of 1,000 flights shows a total of 300 bags were lost. Thus, the arithmetic mean number of lost bags per flight is 0.3 (300/1,000). If the number of lost bags per flight follows a Poisson distribution with u = 0.3, find the probability of not losing any bags.
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LO 6-6
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Poisson Probability Distribution – Table
Recall from the previous illustration that the number of lost bags follows a Poisson distribution with a mean of 0.3. Use Appendix B.5 to find the probability that no bags will be lost on a particular flight. What is the probability no bag will be lost on a particular flight?
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LO 6-6
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More About the Poisson Probability Distribution
- The Poisson probability distribution is always positively skewed and the random variable has no specific upper limit.
- The Poisson distribution for the lost bags illustration, where µ=0.3, is highly skewed.
- As µ becomes larger, the Poisson distribution becomes more symmetrical.
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LO 6-6
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136
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290
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2
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s
s
Continuous Probability Distributions
Chapter 07
McGraw-Hill/Irwin
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
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LEARNING OBJECTIVES
LO 7-1 List the characteristics of the uniform distribution.
LO 7-2 Compute probabilities by using the uniform distribution.
LO 7-3 List the characteristics of the normal probability distribution.
LO 7-4 Convert a normal distribution to the standard normal distribution.
LO 7-5 Find the probability that an observation on a normally distributed random variable is between two values.
LO 7-6 Find probabilities using the Empirical Rule.
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The Uniform Distribution
The uniform probability distribution is perhaps the simplest distribution for a continuous random variable.
This distribution is rectangular in shape and is defined by minimum and maximum values.
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LO 7-1 List the characteristics of the uniform distribution.
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The Uniform Distribution – Mean and Standard Deviation
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LO 7-1
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The Uniform Distribution – Example
Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street and College Drive stop every 30 minutes between 6 A.M. and 11 P.M. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 minutes.
Draw a graph of this distribution.
Show that the area of this uniform distribution is 1.00.
How long will a student “typically” have to wait for a bus? In other words, what is the mean waiting time? What is the standard deviation of the waiting times?
What is the probability a student will wait more than 25 minutes?
What is the probability a student will wait between 10 and 20 minutes?
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LO 7-2 Compute probabilities
using the uniform distribution.
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The Uniform Distribution – Example
1. Graph of this distribution.
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LO 7-2
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The Uniform Distribution – Example
2. Show that the area of this distribution is 1.00.
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LO 7-2
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The Uniform Distribution – Example
3. How long will a student “typically” have to wait for a bus? In other words, what is the mean waiting time?
What is the standard deviation of the waiting times?
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LO 7-2
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The Uniform Distribution – Example
4. What is the probability a student will wait more than 25 minutes?
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LO 7-2
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The Uniform Distribution – Example
5. What is the probability a student will wait between 10 and 20 minutes?
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LO 7-2
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Characteristics of a
Normal Probability Distribution
It is bell-shaped and has a single peak.
It is symmetrical about the mean.
It is asymptotic: The curve gets closer and closer to the X-axis but never actually touches it.
The arithmetic mean, median, and mode are equal
The total area under the curve is 1.00.
The area to the left of the mean = area right of mean = 0.5.
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LO 7-3 List the characteristics of the normal probability distribution.
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The Normal Distribution – Graphically
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LO 7-3
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The Family of Normal Distribution
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Equal Means and Different Standard Deviations
LO 7-3
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The Family of Normal Distribution
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Different Means and Standard Deviations
LO 7-3
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The Family of Normal Distribution
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Different Means and Equal Standard Deviations
LO 7-3
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The Standard Normal
Probability Distribution
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
- It is also called the z distribution.
- A z-value is the signed distance between a selected value, designated X, and the population mean , divided by the population standard deviation, σ.
- The formula is:
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LO 7-4 Convert a normal distribution to the standard normal distribution.
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Areas Under the Normal Curve
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LO 7-4
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The Normal Distribution – Example
The weekly incomes of shift foremen in the glass industry follow the normal probability distribution with a mean of $1,000 and a standard deviation of $100.
What is the z-value for the income, let’s call it X, of a foreman who earns $1,100 per week? For a foreman who earns $900 per week?
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LO 7-5 Find the probability that an observation on a normally distributed random variable is between two values.
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Normal Distribution – Finding Probabilities
In an earlier example, we reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1,000 and a standard deviation of $100.
What is the likelihood of selecting a foreman whose weekly income is between $1,000 and $1,100?
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LO 7-5
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Normal Distribution – Finding Probabilities
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LO 7-5
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Normal Distribution – Finding Probabilities Using the Normal Distribution Table
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LO 7-5
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Finding Areas for z Using Excel
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The Excel function
=NORMDIST(x,Mean,Standard_dev,Cumu)
=NORMDIST(1100,1000,100,true)
generates area (probability) from
Z=1 and below
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LO 7-5
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Normal Distribution – Finding Probabilities (Example 2)
Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100.
What is the probability of selecting a shift foreman in the glass industry whose income is:
Between $790 and $1,000?
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Excel Function: =NORMDIST(1000,1000,100,true)-NORMDIST(790,1000,100,true)
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LO 7-5
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Normal Distribution – Finding Probabilities using the Normal Distribution Table
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LO 7-5
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Normal Distribution – Finding Probabilities (Example 3)
Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100.
What is the probability of selecting a shift foreman in the glass industry whose income is:
Less than $790?
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Excel Function: =NORMDIST(790,1000,100,true)
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LO 7-5
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Normal Distribution – Finding Probabilities Using the Normal Distribution Table
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LO 7-5
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Normal Distribution – Finding Probabilities (Example 4)
Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100.
What is the probability of selecting a shift foreman in the glass industry whose income is:
Between $840 and $1,200?
Excel Function: =NORMSDIST(2.0)-NORMSDIST(-1.6)
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LO 7-5
*
Normal Distribution – Finding Probabilities Using the Normal Distribution Table
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LO 7-5
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Normal Distribution – Finding Probabilities (Example 5)
Refer to the information regarding the weekly income of shift foremen in the glass industry. The distribution of weekly incomes follows the normal probability distribution with a mean of $1,000 and a standard deviation of $100.
What is the probability of selecting a shift foreman in the glass industry whose income is:
Between $1,150 and $1,250
Excel Function: =NORMSDIST(2.5)-NORMSDIST(1.5)
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LO 7-5
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Normal Distribution – Finding Probabilities Using the Normal Distribution Table
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LO 7-5
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Using z in Finding X Given Area – Example
Layton Tire and Rubber Company wishes to set a minimum mileage guarantee on its new MX100 tire. Tests reveal the mean mileage is 67,900 with a standard deviation of 2,050 miles and that the distribution of miles follows the normal probability distribution. Layton wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced.
What minimum guaranteed mileage should Layton announce?
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LO 7-5
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Using z in Finding X Given Area – Example
Set the minimum guaranteed mileage (X) so that no more than 4 percent of the tires will be replaced.
Given Data:
µ = 67,900
σ = 2,050
X = ?
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LO 7-5
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Using z in Finding X Given Area – Example
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LO 7-5
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Using z in Finding X Given Area – Example
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LO 7-5
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Using z in Finding X Given Area – Excel
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LO 7-5
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The Empirical Rule
- About 68 percent of the area under the normal curve is within one standard deviation of the mean.
- About 95 percent is within two standard deviations of the mean.
- Practically all is within three standard deviations of the mean.
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LO 7-6 Find probabilities using the Empirical Rule.
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The Empirical Rule – Example
As part of its quality assurance program, the Autolite Battery Company conducts tests on battery life. For a particular D-cell alkaline battery, the mean life is 19 hours. The useful life of the battery follows a normal distribution with a standard deviation of 1.2 hours.
Answer the following questions.
1. About 68 percent of the batteries failed between what two values?
2. About 95 percent of the batteries failed between what two values?
3. Virtually all of the batteries failed between what two values?
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LO 7-6
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The Empirical Rule – Example
As part of its quality assurance program, the Autolite Battery Company conducts tests on battery life. For a particular D-cell alkaline battery, the mean life is 19 hours. The useful life of the battery follows a normal distribution with a standard deviation of 1.2 hours.
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LO 7-6
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